Closed surfaces with bounds on their Willmore energy
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, p. 605-634

The Willmore energy of a closed surface in n is the integral of its squared mean curvature, and is invariant under Möbius transformations of n . We show that any torus in 3 with energy at most 8π-δ has a representative under the Möbius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ>0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p1 in 3 or 4 , assuming suitable energy bounds which are sharp for n=3. Moreover, the conformal type is controlled in terms of the energy bounds.

Published online : 2019-02-22
Classification:  53A05,  53A30,  53C21,  49Q15
@article{ASNSP_2012_5_11_3_605_0,
     author = {Kuwert, Ernst and Sch\"atzle, Reiner},
     title = {Closed surfaces with bounds on their Willmore energy},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {3},
     year = {2012},
     pages = {605-634},
     zbl = {1260.53027},
     mrnumber = {3059839},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0}
}
Kuwert, Ernst; Schätzle, Reiner. Closed surfaces with bounds on their Willmore energy. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 605-634. http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0/

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